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I have a hard time explaining to my children how addition of fractions work. Not how to do that, but how to feel the intuitive way. And percentages, to some ext

I have a hard time explaining to my children how addition of fractions work. Not how to do that, but how to feel the intuitive way. And percentages, to some extend. The sad part is that I have a PhD in physics and an engineering degree in CS. I used to teach physics at the uni and loved it. My children like me to explain them physics because I love it so much and they appreciate the analogies, their limits etc. It is just these freaking fractions and their convoluted addition I do not know how to properly convey.

ezequiel-garzon 7y ago
To me what is utterly crucial is to use pie pictures to convince children (and adults...) that 1/2 + 1/3 cannot possibly equal 2/5. If they actually buy this that’s a pretty good inflection point in their education on fractions.
- BrandoElFollito 7y ago
  After going though the education of two children who like math very much I realize that fractions on their own are very natural. It is their addition which is not, as it does not have any obvious counterpart in simple nature. I had a math teacher at the university who told us once "there is a neat trick which is normally taught next year but is useless by then and super useful today. Just use it without too much thinking." This is my reasoning too, I showed them the always working way to add fractions (cross multiplication) and hope that someday they will appreciate the beauty of the thing.
  - sethammons 7y ago
    As a former high school math teacher, chances are you did a disservice to them. Many kids never "get" fractions, but do remember cross multiplication! And so they use it for anything with a fraction. 1/3 + 4/5=? CrossMiltipy! 1/3 x 4/5=? CrossMiltipy! 1/3 - 4/5=? CrossMiltipy! 1/3 ÷ 4/5=? CrossMiltipy! 1/3 + ? = 4/5 ... CrossMiltipy? Lots of kids complain about math because there is so much to remember. That's a bad sign because there is so little to remember if you understand the concepts as you can always re-discover forgotten tricks. In similar fashion, as a little kid, I was not allowed Velcro shoes when learning to tie my shoes. After I showed I could master tying them, then I had the option to go back to the easier Velcro option.
abdullahkhalids 7y ago
If you want to add 2/3 + 1/4 (=11/12), take three square sheets of paper. Call the area of each one. 1. Cut the first sheet into 3 equal parts. Take 2 of them (labelled A and B) and place them side by side on the third sheet. 2. Cut the second sheet into 4 equal parts. Take 1 of them (labelled C) and place it on the third sheet next to the other two pieces from part 1. 3. Now, figure out how much area the three pieces cover of the third sheet of paper. If you do it right, you can cut the third sheet of paper into 12 pieces, such that the 11 of these pieces will cover A, B and C exactly. Sometimes you will need a fourth sheet of paper in case the sum is greater than 1. There are many variations of this. You can pour 2/3 of the water from one 100 ml graduated cylinder and 1/4 of water from another 100 ml graduated cylinder into a third one. Or use sand and weight it. If all numbers are divisors of 12, you can use the clock.
jfim 7y ago
Just get a few measuring cups and some water. Half a cup and a quarter cup makes three quarter cups, because really, half a cup is two quarters of a cup.
- Broken_Hippo 7y ago
  Which is all well and good... Until you need a half of a quarter (which I'd use tablespoons for as there are 16 tablespoons in a cup), a half of a third of a cup, or need to add 1/3 and 3/4 together. Which isn't so bad for a recipe because most things aren't that precise in cooking - but not so easy for all students to work out, especially without having a tablespoon to use. This is even so when you have tablespoons: Those 1/3 cups don't divide neatly into tablespoons. That said, it isn't like you cannot illustrate it, of course, but just that this method isn't the best. I've just memorised the cooking stuff since I only have a 2-person household.
  - dragonwriter 7y ago
    > Those 1/3 cups don't divide neatly into tablespoons. But they do into teaspoons.
    - Broken_Hippo 7y ago
      Sure, but if you actually count out 16 teaspoons, you are likely to lose count :)
Tharkun 7y ago
When I was first taught fractions in primary school, the teacher had a bunch of rods of equal length. Each rod was divided into segments of different colours. Each segment could be "fractured" off the rod because they were attached by magnets. I think they went from 1 to 10 or so segments. This allowed for a great visual (and tactile) illustration of how fractions work. 1/2 + 1/4? No worries, just break off some pieces and add them together. Then you can compare the length to various other configurations, and you can viscerally experience that 3/4 really does equal 6/8. I don't know whether this is a standard way of teaching fractions, or whether this teacher was particularly motivated or whatever. But it made a lasting impression on me. Thanks, miss Annie.
- pharke 7y ago
  I believe they are called Cuisenaire Rods https://en.wikipedia.org/wiki/Cuisenaire_rods https://en.wikipedia.org/wiki/Cuisenaire_rods
rs23296008n1 7y ago
Pizza slices. Different sized slices. Fractions of a pizza. Half a pizza plus a quarter plus a quarter = one pizza. A pizza is 4/4 of a pizza = a pizza sliced into 4 but still together. The benefit of this approach is you can easily go to 1/8 and 1/16 and do addition and subtraction. Even multiplication and division. Get two pizzas to show 8/4 and 16/8 etc. And you can eat the whole thing afterwards. Fractions are delicious.
matwood 7y ago
> I have a hard time explaining to my children how addition of fractions work. The best way I found to learn fractions was through carpentry. Measuring and cutting over and over quickly builds the intuitive sense of how fractions work.

ezequiel-garzon 7y ago

To me what is utterly crucial is to use pie pictures to convince children (and adults...) that 1/2 + 1/3 cannot possibly equal 2/5. If they actually buy this that’s a pretty good inflection point in their education on fractions.

BrandoElFollito 7y ago
After going though the education of two children who like math very much I realize that fractions on their own are very natural. It is their addition which is not, as it does not have any obvious counterpart in simple nature. I had a math teacher at the university who told us once "there is a neat trick which is normally taught next year but is useless by then and super useful today. Just use it without too much thinking." This is my reasoning too, I showed them the always working way to add fractions (cross multiplication) and hope that someday they will appreciate the beauty of the thing.
- sethammons 7y ago
  As a former high school math teacher, chances are you did a disservice to them. Many kids never "get" fractions, but do remember cross multiplication! And so they use it for anything with a fraction. 1/3 + 4/5=? CrossMiltipy! 1/3 x 4/5=? CrossMiltipy! 1/3 - 4/5=? CrossMiltipy! 1/3 ÷ 4/5=? CrossMiltipy! 1/3 + ? = 4/5 ... CrossMiltipy? Lots of kids complain about math because there is so much to remember. That's a bad sign because there is so little to remember if you understand the concepts as you can always re-discover forgotten tricks. In similar fashion, as a little kid, I was not allowed Velcro shoes when learning to tie my shoes. After I showed I could master tying them, then I had the option to go back to the easier Velcro option.

abdullahkhalids 7y ago

If you want to add 2/3 + 1/4 (=11/12), take three square sheets of paper. Call the area of each one. 1. Cut the first sheet into 3 equal parts. Take 2 of them (labelled A and B) and place them side by side on the third sheet. 2. Cut the second sheet into 4 equal parts. Take 1 of them (labelled C) and place it on the third sheet next to the other two pieces from part 1. 3. Now, figure out how much area the three pieces cover of the third sheet of paper. If you do it right, you can cut the third sheet of paper into 12 pieces, such that the 11 of these pieces will cover A, B and C exactly. Sometimes you will need a fourth sheet of paper in case the sum is greater than 1. There are many variations of this. You can pour 2/3 of the water from one 100 ml graduated cylinder and 1/4 of water from another 100 ml graduated cylinder into a third one. Or use sand and weight it. If all numbers are divisors of 12, you can use the clock.

jfim 7y ago

Just get a few measuring cups and some water. Half a cup and a quarter cup makes three quarter cups, because really, half a cup is two quarters of a cup.

Broken_Hippo 7y ago
Which is all well and good... Until you need a half of a quarter (which I'd use tablespoons for as there are 16 tablespoons in a cup), a half of a third of a cup, or need to add 1/3 and 3/4 together. Which isn't so bad for a recipe because most things aren't that precise in cooking - but not so easy for all students to work out, especially without having a tablespoon to use. This is even so when you have tablespoons: Those 1/3 cups don't divide neatly into tablespoons. That said, it isn't like you cannot illustrate it, of course, but just that this method isn't the best. I've just memorised the cooking stuff since I only have a 2-person household.
- dragonwriter 7y ago
  > Those 1/3 cups don't divide neatly into tablespoons. But they do into teaspoons.
  - Broken_Hippo 7y ago
    Sure, but if you actually count out 16 teaspoons, you are likely to lose count :)

Tharkun 7y ago

When I was first taught fractions in primary school, the teacher had a bunch of rods of equal length. Each rod was divided into segments of different colours. Each segment could be "fractured" off the rod because they were attached by magnets. I think they went from 1 to 10 or so segments. This allowed for a great visual (and tactile) illustration of how fractions work. 1/2 + 1/4? No worries, just break off some pieces and add them together. Then you can compare the length to various other configurations, and you can viscerally experience that 3/4 really does equal 6/8. I don't know whether this is a standard way of teaching fractions, or whether this teacher was particularly motivated or whatever. But it made a lasting impression on me. Thanks, miss Annie.

pharke 7y ago
I believe they are called Cuisenaire Rods https://en.wikipedia.org/wiki/Cuisenaire_rods https://en.wikipedia.org/wiki/Cuisenaire_rods

rs23296008n1 7y ago

Pizza slices. Different sized slices. Fractions of a pizza. Half a pizza plus a quarter plus a quarter = one pizza. A pizza is 4/4 of a pizza = a pizza sliced into 4 but still together. The benefit of this approach is you can easily go to 1/8 and 1/16 and do addition and subtraction. Even multiplication and division. Get two pizzas to show 8/4 and 16/8 etc. And you can eat the whole thing afterwards. Fractions are delicious.

matwood 7y ago

> I have a hard time explaining to my children how addition of fractions work. The best way I found to learn fractions was through carpentry. Measuring and cutting over and over quickly builds the intuitive sense of how fractions work.